Let $X,Y$ and $Z$ be Banach spaces, and let $\prod_p(Y,Z) (1\leq p<\infty)$ denote the space of $p$-summing operators from $Y$ to $Z$. We show that, if $X$ is a {\it \$}$_\infty$-space, then a bounded linear operator $T: X\hat \otimes_εY\longrightarrow Z$ is 1-summing if and only if a naturally associated operator $T^#: X\longrightarrow \prod_1(Y,Z)$ is 1-summing. This result need not be true if $X$ is not a {\it \$}$_\infty$-space. For $p>1$, several examples are given with $X=C[0,1]$ to show that $T^#$ can be $p$-summing without $T$ being $p$-summing. Indeed, there is an operator $T$ on $C[0,1]\hat \otimes_ε\ell_1$ whose associated operator $T^#$ is 2-summing, but for all $N\in \N$, there exists an $N$-dimensional subspace $U$ of $C[0,1]\hat \otimes_ε\ell_1$ such that $T$ restricted to $U$ is equivalent to the identity operator on $\ell^N_\infty$. Finally, we show that there is a compact Hausdorff space $K$ and a bounded linear operator $T:\ C(K)\hat \otimes_ε\ell_1\longrightarrow \ell_2$ for which $T^#:\ C(K)\longrightarrow \prod_1(\ell_1, \ell_2)$ is not 2-summing.
Grothendieck's theorem asserts that every continuous linear operator from $\ell_1$ to $\ell_2$ is absolutely $(1,1)$-summing. This kind of result is commonly called coincidence result. In this paper we investigate coincidence results in the multilinear setting, showing how the cotype of the spaces involved affect such results. The special role played by $\ell_1$ spaces is also investigated with relation to interpolation of tensor products. In particular, an open problem on the interpolation of $m$ injective tensor products is solved.
We relate the injectivity of the canonical map from $C(B_{E'})$ to $L_p(μ)$, where $μ$ is a regular Borel probability measure on the closed unit ball $B_{E'}$ of the dual $E'$ of a Banach space $E$ endowed with the weak* topology, to the existence of injective $p$-summing linear operators/$p$-dominated homogeneous polynomials defined on $E$ having $μ$ as a Pietsch measure. As an application we fill the gap in the proofs of some results of concerning Pietsch-type factorization of dominated polynomials.
In this article, we study the ideals of mid $p$-summing operators. We obtain representation of these operator ideals by tensor norms. These tensor norms are defined by using a particular kind of sequential dual of the class of mid $p$-summable sequences. As a consequence, we prove a characterization of the adjoints of weakly and absolutely mid $p$-summing operators in terms of the operators that are defined by the transformation of dual spaces of certain vector-valued sequence spaces.
In addition to Pisier's counterexample of a non-accessible maximal Banach ideal, we will give a large class of maximal Banach ideals which {\it{are accessible}}. The first step is implied by the observation that a "good behaviour" of trace duality, which is canonically induced by conjugate operator ideals can be extended to adjoint Banach ideals, if and only if these adjoint ideals satisfy an accessibility condition (theorem 3.1). This observation leads in a natural way to a characterization of accessible injective Banach ideals, where we also recognize the appearance of the ideal of {\it{abso…
We give conditions that ensure that an operator satisfying a Piestch domination in a given setting also satisfies a Piestch domination in a different setting. From this we derive that a bounded mutlilinear operator $T$ is Lipschitz $p$-summing if and only if the mapping $ f_T(x_1\otimes\cdots \otimes x_n):=T(x_1,\ldots, x_n)$ is Lipschitz $p$-summing. The results are based on the projective tensor norm. An example with the Hilbert tensor norm is provided to show that the statement may not hold when a reasonable cross-norm other than the projective tensor norm is considered.
In this paper, we show that four main diagonal spaces of injective tensor products are pairwise isometrically isomorphic. When E is a Banach lattice, we show that the tensor diagonal of E is a 1-unconditional basic sequence in both the n-fold injective tensor product of E and the n-fold symmetric injective tensor product of E.
We apply the geometric approach provided by $Σ$-operators to develop a theory of $p$-summability for multilinear operators. In this way, we introduce the notion of Lipschitz $p$-summing multilinear operators and show that it is consistent with a general panorama of generalization: Namely, they satisfy Pietsch-type domination and factorization theorems and generalizations of the inclusion Theorem, Grothendieck's coincidence Theorems, the weak Dvoretsky-Rogers Theorem and a Lindenstrauss-Pelczyńsky Theorem. We also characterize this new class in tensorial terms by means of a Chevet-Saphar-type t…
We prove when a Banach ideal of linear operators defined, or characterized, by the transformation of vector-valued sequences is maximal. Known results are recovered as particular cases and new information is obtained. To accomplish this task we study a tensor quasi-norm determined by the underlying sequence classes. The duality theory for these tensor quasi-norms is also developed.
In this paper we prove that the $3$-fold injective tensor product $\ell_1 \widehat{\otimes}_\varepsilon \ell_1 \widehat{\otimes}_\varepsilon \ell_1 $ is not isomorphic to any subspace of $\ell_1 \widehat{\otimes}_\varepsilon \ell_1$. This result provides a new solution to a problem of Diestel on the projective tensor products of $c_0.$ Moreover, this result implies that for any infinite countable compact space $K,$ the $3$-fold projective tensor product $C(K) \widehat{\otimes}_πC(K)\widehat{\otimes}_πC(K)$ is not isomorphic to any quotient of $C(K) \widehat{\otimes}_πC(K)$.
Given any Banach space $X$ and any weak*-compact subset $K$ of $X^*$, we compute the Szlenk index of the weak*-closed, convex hull of $K$ as a function of the Szlenk index of $K$. Also as an application, we compute the Szlenk index of any injective tensor product of two operators. In particular, we compute the Szlenk index of an injective tensor product $\tens$ in terms of $Sz(X)$ and $Sz(Y)$. As another application, we give a complete characterization of those ordinals which occur as the Szlenk index of a Banach space, as well as those ordinals which occur as the Bourgain $\ell_1$ or $c_0$ in…
The aim of this note is to study some geometrical properties like diameter two properties, octahedrality and almost squareness in the setting of (symmetric) tensor product spaces. In particular, we show that the injective tensor product of two octahedral Banach spaces is always octahedral, the injective tensor product of an almost square Banach space with any Banach space is almost square, and the injective symmetric tensor product of an octahedral Banach space is octahedral.
The tensor product of two ordered vector spaces can be ordered in more than one way, just as the tensor product of normed spaces can be normed in multiple ways. Two natural orderings have received considerable attention in the past, namely the ones given by the projective and injective (or biprojective) cones. This paper aims to show that these two cones behave similarly to their normed counterparts, and furthermore extends the study of these two cones from the algebraic tensor product to completed locally convex tensor products. The main results in this paper are the following: (i) drawing pa…
We prove a general result on the factorization of matrix-valued analytic functions. We deduce that if $(E_0,E_1)$ and $(F_0,F_1)$ are interpolation pairs with dense intersections, then under some conditions on the spaces $E_0$, $E_1$, $F_0$ and $F_1$, we have $$ [E_0\hat\otimes F_0,E_1\hat\otimes F_1]_t= [E_0, E_1]_t\hat\otimes[F_0, F_1]_t, 0 < t< 1.$$ We find also conditions on the spaces $E_0$, $E_1$, $F_0$ and $F_1$, so that the following holds $$ [E_0\wcheck\otimes F_0,E_1\wcheck\otimes F_1]_t= [E_0,E_1]_t\wcheck\otimes [F_0,F_1]_t, 0 <t< 1.$$ Some applications of these results are also co…
Tensor products of convex cones have recently come up in different areas, ranging from functional analysis and operator theory to approximation theory and theoretical physics. However, most of the existing literature focuses either on Archimedean lattice cones or on closed proper cones in finite-dimensional spaces, thereby excluding many cones, including even standard cones such as an infinite-dimensional positive semidefinite cone. For general cones, results are few and far between, and many basic questions remain unanswered. In this memoir, we develop the theory of tensor products of convex …
Let $E$ and $F$ be complex Banach spaces, $U$ be an open subset of $E$ and $1\leq p\leq\infty$. We introduce and study the notion of a Cohen strongly $p$-summing holomorphic mapping from $U$ to $F$, a holomorphic version of a strongly $p$-summing linear operator. For such mappings, we establish both Pietsch domination/factorization theorems and analyse their linearizations from $\mathcal{G}^\infty(U)$ (the canonical predual of $\mathcal{H}^\infty(U)$) and their transpositions on $\mathcal{H}^\infty(U)$. Concerning the space $\mathcal{D}_p^{\mathcal{H}^\infty}(U,F)$ formed by such mappings and …
The main purpose of this paper is the study of a~new class of summing multilinear operators acting from the product of Banach lattices with some nontrivial lattice convexity. A~mixed Pietsch-Maurey-Rosenthal type factorization theorem for these operators is proved under weaker convexity requirements than the ones that are needed in the Maurey-Rosenthal factorization through products of $L^q$-spaces. A~by-product of our factorization is an extension of multilinear operators defined by a~$q$-concavity type property to a~product of special Banach function lattices which inherit some lattice-geome…
We study in this paper some property of Lipschitz mappings which admit factorization through an operator ideal. We try to construct Lipschitz cross-norms from known tensor norms in order to represent certain classes of Lipschitz mappings. Inspired by the definition of p-summing linear operators we introduce a new concpet in the the category of Lipschitz mappings that is called strictly Lipschitz p-summing.
We study the space of $p$-compact operators $\mathcal K_p$, using the theory of tensor norms and operator ideals. We prove that $\mathcal K_p$ is associated to $/d_p$, the left injective associate of the Chevet-Saphar tensor norm $d_p$ (which is equal to $g_{p'}'$). This allows us to relate the theory of $p$-summing operators with that of $p$-compact operators. With the results known for the former class and appropriate hypothesis on $E$ and $F$ we prove that $\mathcal K_p(E;F)$ is equal to $\mathcal K_q(E;F)$ for a wide range of values of $p$ and $q$, and show that our results are sharp. We a…
The aim of this paper is two-fold. Firstly we provide necessary and sufficient criteria for the existence of a strict deformation quantization of algebraic tensor products of Poisson algebras, and secondly, we discuss the existence of products of KMS states. As an application, we discuss the correspondence between quantum and classical Hamiltonians in spin systems and we provide a relation between the resolvent of Schödinger operators for non-interacting many-particle systems and quantization maps.
We continue the investigation of the behaviour of diameter two properties in tensor products of Banach spaces. Our main result shows that the symmetric strong diameter two property is stable by taking projective tensor products. We also prove a result for the symmetric strong diameter two property for the injective tensor product.
We prove a complex interpolation formula for the injective tensor product of vector-valued Banach function spaces satisfying certain geometric assumptions. This result unifies results of Kouba, and moreover, our approach offers an alternate proof of Kouba's interpolation formula for scalar-valued Banach function spaces.
We prove a formula for the Szlenk power type of the injective tensor product of Banach spaces with Szlenk index at most $ω$. We also show that the Szlenk power type as well as summability of the Szlenk index are separably determined, and we extend some of our recent results concerning direct sums.
We show that the numerical index of any operator ideal is less than or equal to the minimum of the numerical indices of the domain and the range. Further, we show that the numerical index of the ideal of compact operators or the ideal of weakly compact operators is less than or equal to the numerical index of the dual of the domain, and this result provides interesting examples. We also show that the numerical index of a projective or injective tensor product of Banach spaces is less than or equal to the numerical index of any of the factors. Finally, we show that if a projective tensor produc…
In this work, we study non-rough norms in L(X,Y), the space of bounded linear operators between Banach spaces X and Y. We prove that L(X,Y) has non-rough norm if and only if X* and Y have non-rough norm. We show that the injective tensor product of X and Y has non-rough norm if and only if both X and Y have non-rough norm. We also give an example to show that non-rough norms are not stable under projective tensor product. We also study a related concept namely the small diameter properties in the context of L(X,Y)*. These results leads to a discussion on stability of the small diameter propert…
Inspired by ideas of R. Schatten in his celebrated monograph on a theory of cross-spaces, we introduce the notion of a Lipschitz tensor product X\boxtimes E of a pointed metric space and a Banach space E as a certain linear subspace of the algebraic dual of Lipo(X,E^*). We prove that <Lipo(X,E^*),X\boxtimes E> forms a dual pair. We prove that X\boxtimes E is linearly isomorphic to the linear space of all finite-rank continuous linear operators from (X^#,T) into E, where X^# denotes the space Lipo(X,K) and T is the topology of pointwise convergence of X^#. The concept of Lipschitz tensor produc…
In this paper we show how \emph{the metric theory of tensor products} developed by Grothendieck perfectly fits in the study of channel capacities, a central topic in \emph{Shannon's information theory}. Furthermore, in the last years Shannon's theory has been generalized to the quantum setting to let the \emph{quantum information theory} step in. In this paper we consider the classical capacity of quantum channels with restricted assisted entanglement. In particular these capacities include the classical capacity and the unlimited entanglement-assisted classical capacity of a quantum channel. …
We continue the investigation of the behaviour of octahedral norms in tensor products of Banach spaces. Firstly, we will prove the existence of a Banach space $Y$ such that the injective tensor products $l_1\widehat{\otimes}_\varepsilon Y$ and $L_1\widehat{\otimes}_\varepsilon Y$ both fail to have an octahedral norm, which solves two open problems from the literature. Secondly, we will show that in the presence of the metric approximation property octahedrality is preserved from a non-reflexive $L$-embedded Banach space taking projective tensor products with an arbitrary Banach space.
We study the asymptotic behaviour of large tensor powers of normed spaces and of operators between them. We define the tensor radius of a finite-dimensional normed space $X$ as the limit of the sequence $A_k^{1/k}$, where $A_k$ is the equivalence constant between the projective and injective norms on $X^{\otimes k}$. We show that Euclidean spaces are characterized by the property that their tensor radius equals their dimension. Moreover, we compute the tensor radius for spaces with enough symmetries, such as the spaces $\ell_p^n$. We also define the tensor radius of an operator $T$ as the limi…
We study tensor norms that destroy unconditionality in the following sense: for every Banach space $E$ with unconditional basis, the $n$-fold tensor product of $E$ (with the corresponding tensor norm) does not have unconditional basis. We establish an easy criterion to check weather a tensor norm destroys unconditionality or not. Using this test we get that all injective and projective tensor norms different from $\varepsilon$ and $π$ destroy unconditionality, both in full and symmetric tensor products. We present applications to polynomial ideals: we show that many usual polynomial ideals nev…
We show that, in some cases, the projective and the injective tensor products of two Banach spaces do not have the Dunford-Pettis property (DPP). As a consequence, we obtain that $(c_0\hat{\otimes}_πc_0)^{**}$ fails the DPP. Since $(c_0\hat{\otimes}_πc_0)^{*}$ does enjoy it, this provides a new space with the DPP whose dual fails to have it. We also prove that, if $E$ and $F$ are ${\mathscr L}_1$-spaces, then $E\hat{\otimes}_εF$ has the DPP if and only if both $E$ and $F$ have the Schur property. Other results and examples are given.
Several new characterizations of Banach spaces containing a subspace isomorphic to $\ell^1$, are obtained. These are applied to the question of when $\ell^1$ embeds in the injective tensor product of two Banach spaces.
Given two Banach spaces $X$ and $Y$, we analyze when the projective tensor product $X\widehat{\otimes}_πY$ has Corson's property (C) or is weakly Lindelöf determined (WLD), subspace of a weakly compactly generated (WCG) space or subspace of a Hilbert generated space. For instance, we show that: (i) $X\widehat{\otimes}_πY$ is WLD if and only if both $X$ and $Y$ are WLD and all operators from $X$ to $Y^*$ and from $Y$ to $X^*$ have separable range; (ii) $X\widehat{\otimes}_πY$ is subspace of a WCG space if the same holds for both $X$ and $Y$ under the assumption that every operator from $X$ to $…
Motivated by the theory of tensor algebras and multivariable C*-dynamics, we revisit two fundamental techniques in the theory of C*-correspondences, the "addition of a tail" to a non-injective C*-correspondence and the dilation of an injective C*-correspondence to an essential Hilbert bimodule. We provide a very broad scheme for "adding a tail" to a non-injective C*-correspondence; our scheme includes the "tail" of Muhly and Tomforde as a special case. We illustrate the diversity and necessity of our tails with several examples from the theory of multivariable C*-dynamics. We also exhibit a tr…
We prove two identities that connect some natural tensor products in the category $\sf{LCS}$ of locally convex spaces with the tensor products in the category $\sf{Ste}$ of stereotype spaces. In particular, we give sufficient conditions under which the identity $$ X^\vartriangle\odot Y^\vartriangle\cong (X^\vartriangle\cdot Y^\vartriangle)^\vartriangle\cong (X\cdot Y)^\vartriangle $$ holds, where $\odot$ is the injective tensor product in the category $\sf{Ste}$, $\cdot$, the primary tensor product in the category $\sf{LCS}$, and $\vartriangle$, the pseudosaturation operation in the category $…
We use the Aron-Berner extension to prove that the set of extreme points of the unit ball of the space of integral polynomials over a real Banach space $X$ is $\{\pm ϕ^k: ϕ\in X^*, \| ϕ\|=1\}$. With this description we show that, for real Banach spaces $X$ and $Y$, if $X$ is a non trivial $M$-ideal in $Y$, then $\hat\bigotimes^{k,s}_{ε_{k,s}} X$ (the $k$-th symmetric tensor product of $X$ endowed with the injective symmetric tensor norm) is \emph{never} an $M$-ideal in $\hat\bigotimes^{k,s}_{ε_{k,s}} Y$. This result marks up a difference with the behavior of non-symmetric tensors since, when $…
The injective tensor product of normal representable bimodules over von Neumann algebras is shown to be normal. The usual Banach module projective tensor product of central representable bimodules over an Abelian C$^*$-algebra is shown to be representable. A normal version of the projective tensor product is introduced for central normal bimodules.
We define and study XOR games in the framework of general probabilistic theories, which encompasses all physical models whose predictive power obeys minimal requirements. The bias of an XOR game under local or global strategies is shown to be given by a certain injective or projective tensor norm, respectively. The intrinsic (i.e.\ model-independent) advantage of global over local strategies is thus connected to a universal function $r(n,m)$ called 'projective-injective ratio'. This is defined as the minimal constant $ρ$ such that $\|\cdot\|_{X\otimes_πY}\leqρ\,\|\cdot\|_{X\otimes_\varepsilon …
We show that the symmetric injective tensor product space $\hat{\otimes}_{n,s,ε}E$ is not complex strictly convex if E is a complex Banach space of $\dim E \ge 2$ and if $n\ge 2$ holds. It is also reproved that $\ell_\infty$ is finitely represented in $\hat{\otimes}_{n,s,ε}E$ if E is infinite dimensional and if $n\ge 2$ holds, which was proved in the other way by Dineen.
We examine some structural properties of (injective and projective) tensor products of $\ell_p$-spaces (projections, complemented subspaces, reflexivity, isomorphisms, etc.). We combine these results with combinatorial arguments to address the question of primarity for these spaces and their duals. Our main results are: \medbreak \item{(1)} If $1<p<\infty$, then $B(\ell_p)\approx B(L_p)$ ($B(X)$ consists of the bounded linear operators on $X$). \medbreak \item{(2)} If ${1\over p_i}+{1\over p_j}\leq1$ for every $i\neq j$, or if all of the $p_i$'s are equal, then $\ell_{p_1}\hat{\otimes}\cdots \…
We characterize those bounded multilinear operators that factor through a Hilbert space in terms of its behavior in finite sequences. This extends a result, essentially due to S. Kwapień, from the linear to the multilinear setting. We prove that Hilbert-Schmidt and Lipschitz $2$-summing multilinear operators naturally factor through a Hilbert space. It is also proved that the class $Γ$ of all multilinear operators that factor through a Hilbert space is a maximal multi-ideal; moreover, we give an explicit formulation of a finitely generated tensor norm $γ$ which is in duality with $Γ$.
We define the $k$:th moment of a Banach space valued random variable as the expectation of its $k$:th tensor power; thus the moment (if it exists) is an element of a tensor power of the original Banach space. We study both the projective and injective tensor products, and their relation. Moreover, in order to be general and flexible, we study three different types of expectations: Bochner integrals, Pettis integrals and Dunford integrals. One of the problems studied is whether two random variables with the same injective moments (of a given order) necessarily have the same projective moments; …
In this paper, we introduce the concept of a pseudo weakly compact operator of order $ p $ between Banach spaces. Also we study the notion of $ p $-Dunford-Pettis relatively compact property which is in "general" weaker than the Dunford-Pettis relatively compact property and gives some characterizations of Banach spaces which have this property. Moreover, by using the notion of $ p $-Right subsets of a dual Banach space, we study the concepts of $ p $-sequentially Right and weak $ p $-sequentially Right properties on Banach spaces. Furthermore, we obtain some suitable conditions on Banach spac…
Given operator spaces $V$ and $W$, let $\widetilde{W}$ denote the opposite operator space structure on the same underlying Banach space. Although the identity map $W\to \widetilde{W}$ is in general not completely bounded, we show that the identity map on $V\otimes W$ extends to a contractive linear map $V\widehat{\otimes} W \to V\otimes_{\min} \widetilde{W}$, where $\widehat{\otimes}$ and $\otimes_{\rm min}$ denote the projective and injective tensor products of operator spaces. In future work, this will be applied to construct antisymmetric $2$-cocycles on certain Fourier algebras.
In a preceding work it is determined when a centrally symmetric convex body in $\mathbb{R}^d,$ $d=d_1\cdots d_l,$ is the closed unit ball of a reasonable crossnorm on $\mathbb{R}^{d_1}\otimes\cdots\otimes\mathbb{R}^{d_l}.$ Consequently, the class of tensorial bodies is introduced, an associated tensorial Banach-Mazur distance is defined and the corresponding Banach-Mazur type compactum is proved to exist. In this paper, we introduce the hyperspace of these convex bodies. We called "the space of tensorial bodies". It is proved that the group of linear isomorphisms on $\mathbb{R}^{d_1}\otimes\cd…
In this work, we investigate measurement incompatibility in general probabilistic theories (GPTs). We show several equivalent characterizations of compatible measurements. The first is in terms of the positivity of associated maps. The second relates compatibility to the inclusion of certain generalized spectrahedra. For this, we extend the theory of free spectrahedra to ordered vector spaces. The third characterization connects the compatibility of dichotomic measurements to the ratio of tensor crossnorms of Banach spaces. We use these characterizations to study the amount of incompatibility …
